Subgroup and quotient

Algebra Level 4

Let \(G\) be a group, and \(N\) a normal subgroup. Which of the following statements is/are always true?

I. If \(N\) is finite and \(G/N\) is finite, then \(G\) is finite.
II. If \(N\) is finite and cyclic and \(G/N\) is finite and cyclic, then \(G\) is finite and cyclic.
III. If \(N\) is abelian and \(G/N\) is abelian, then \(G\) is abelian.

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Notation:

  • A finite cyclic group is a group that is isomorphic to \( {\mathbb Z}_n,\) the integers mod \(n,\) for some \(n.\)
  • An abelian group is a group whose operation is commutative: \(x * y = y * x\) for all \(x,y \in G.\)
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